论文标题
单一群体的单词度量：改进的小范围
Word Measures on Unitary Groups: Improved Bounds for Small Representations
论文作者
论文摘要
让$ f $是一组免费的等级$ r $，并在f $中修复了一些$ w \。对于任何紧凑的组$ g $，我们都可以定义$μ__{w，g} $ by $ g $ by（haar-）均匀采样$ g_1，...，g_r \ in G $中的g_r \ in g $并评估$ w（g_1，...，...，g_r）$。在[Arxiv：1802.04862]中，Magee和Puder研究了$ g $是unital unital $ u（n）$的案例，并分析了$μ__{w，u（n）} $的矩量如何以$ n $为单位。特别是，他们在这些时刻获得了渐近范围，与换向器长度和稳定的换向器长度有关。我们继续他们的工作线，对$μ__{w，u（n）} $的矩形行为进行更精确的分析，这表明它与$ w $的另一代代数不变性有关：其原始等级。 此外，我们证明了一种猜想的汉尼和puder的特殊情况（[[Arxiv：2009.00897，猜想1.13]），内容涉及$μ__{W，U（n）} $下的不可减至特征的预期值的渐近行为。
Let $F$ be a free group of rank $r$ and fix some $w\in F$. For any compact group $G$ we can define a measure $μ_{w,G}$ on $G$ by (Haar-)uniformly sampling $g_1,...,g_r\in G$ and evaluating $w(g_1,...,g_r)$. In [arXiv:1802.04862], Magee and Puder studied the case where $G$ is the unitary group $U(n)$, and analyzed how the moments of $μ_{w,U(n)}$ behave as a function of $n$. In particular, they obtained asymptotic bounds on those moments, related to the commutator length and the stable commutator length of $w$. We continue their line of work and give a more precise analysis of the asymptotic behavior of the moments of $μ_{w, U(n)}$, showing that it is related to another algebraic invariant of $w$: its primitivity rank. In addition, we prove a special case of a conjecture of Hanany and Puder ([arXiv:2009.00897, Conjecture 1.13]) regarding the asymptotic behaviour of expected values of irreducible characters under $μ_{w, U(n)}$.
